Given a subgroup $~$H$~$ of Group $~$G$~$, the *left cosets* of $~$H$~$ in $~$G$~$ are sets of the form $~$\{ gh : h \in H \}$~$, for some $~$g \in G$~$.
This is written $~$gH$~$ as a shorthand.

Similarly, the *right cosets* are the sets of the form $~$Hg = \{ hg: h \in H \}$~$.

# Examples

%%%knows-requisite(Symmetric group):

## Symmetric group

In $~$S_3$~$, the Symmetric group on three elements, we can list the elements as $~$\{ e, (123), (132), (12), (13), (23) \}$~$, using cycle notation. Define $~$A_3$~$ (which happens to have a name: the Alternating group) to be the subgroup with elements $~$\{ e, (123), (132) \}$~$.

Then the coset $~$(12) A_3$~$ has elements $~$\{ (12), (12)(123), (12)(132) \}$~$, which is simplified to $~$\{ (12), (23), (13) \}$~$.

The coset $~$(123)A_3$~$ is simply $~$A_3$~$, because $~$A_3$~$ is a subgroup so is closed under the group operation. $~$(123)$~$ is already in $~$A_3$~$. %%%

[todo: more examples, with different requirements]

# Properties

- The left cosets of $~$H$~$ in $~$G$~$ [set_partition partition] $~$G$~$. (Proof.)
- For any pair of left cosets of $~$H$~$, there is a bijection between them; that is, all the cosets are all the same size. (Proof.)

# Why are we interested in cosets?

Under certain conditions (namely that the subgroup $~$H$~$ must be normal), we may define the Quotient group, a very important concept; see the page on "left cosets partition the parent group" for a glance at why this is useful. [todo: there must be a less clumsy way to do it]

Additionally, there is a key theorem whose usual proof considers cosets (Lagrange's theorem) which strongly restricts the possible sizes of subgroups of $~$G$~$, and which itself is enough to classify all the groups of order $~$p$~$ for $~$p$~$ prime. Lagrange's theorem also has very common applications in [-number_theory], in the form of the [fermat_euler_theorem Fermat-Euler theorem].